Theorem subandd-P3.42c.RC 344

Description: Inference Form of subandd-P3.42c 343. †

Hypotheses

Ref	Expression
subandd-P3.42c.RC.1	⊢ (𝜑 ↔ 𝜓)
subandd-P3.42c.RC.2	⊢ (𝜒 ↔ 𝜗)

Assertion

Ref	Expression
subandd-P3.42c.RC	⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗))

Proof of Theorem subandd-P3.42c.RC

Step	Hyp	Ref	Expression
1		subandd-P3.42c.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		subandd-P3.42c.RC.2	. . . 4 ⊢ (𝜒 ↔ 𝜗)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜗))
5	2, 4	subandd-P3.42c 343	. 2 ⊢ (⊤ → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗)))
6	5	ndtruee-P3.18 183	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜗))

Syntax hints:   ↔ wff-bi 104   ∧ wff-and 132  ⊤wff-true 153

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by:  oroverbi-P4.28b  469  imoverbi-P4.30b  479  biasandor-P4.34a  491  psubsplitelof-P6  801